Method, computer program, and storage medium for estimating randomness of function of representative value of random variable by the use of gradient of same function

ABSTRACT

A method of estimating a measure of randomness of a function of at least one representative value of at least one random variable is constructed to have the steps of obtaining the at least one random variable; determining the at least one representative value of the obtained at least one random variable; determining a first statistic of the obtained at least one random variable; determining a gradient of the function with respect to the determined at least one representative value; and transforming the obtained first statistic into a second statistic of the function, using the determined gradient. The step of transforming may be adapted to transform the first statistic into the second statistic, such that the second statistic responds to the first statistic more sensitively in the case of the gradient being steep than in the case of the gradient being gentle.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] Not Applicable.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to a statistical analysis of data,specifically to a technology for estimating a measure of randomness of afunction of at least one random variable.

[0004] 2. Discussion of the Related Art

[0005] Frequently, there are performance measures of systems, which arebased on the means of random variables. For example, a percentage oftime of machine under repair is a function of the mean repair timedivided by the mean time between the beginning of repairs. It isimportant to distinguish between the mean of a function of at least onerandom variable and the function of the means of at least one randomvariable. In the case of the machine repair, it would be possible todivide the individual repair times by the individual times between thebeginning of repairs, and to obtain the mean of this ratio. However,this mean of the function would differ from the function of the means.Only the function of the means represents the correct percentage of themachine under repair.

[0006] Frequently, the means of the random variables are not knownexactly, but rather are based on a set of collected data. Therefore,these means may differ from the true means. Subsequently, the functionof the means may differ from the function of the true means. Frequently,there is interest in a measurement of the accuracy of the function ofthe means. This measurement of accuracy is usually expressed as aconfidence interval around the mean or median, but may also be expressedas a variance, a standard deviation, or a quantile. While thecalculation of such measures is well known in statistical analysis forindividual random variables, it is more difficult for functions of themeans.

[0007] Common uses of the function of at least one mean are frequenciesof occurrences, where the mean frequency is the inverse of the mean timebetween occurrences. Another common uses are percentages of times, wherethe mean percentage is the mean duration divided by the mean timebetween the start of duration's cycles.

[0008] One conventional method to calculate the confidence interval ofthe function of means is called batching, also known as non-overlappingbatch means method. In this method, the sufficiently large sets of dataare split into a number of subsets. The means for each subset iscalculated and subsequently the function of the means is calculated foreach subset. A confidence interval can be constructed on the differentvalues of the function of means.

[0009] However, this conventional method is suitable only forsufficiently large sets of data in order to satisfy the central limittheorem. This method can therefore not be used on small data sets. Inaddition, the confidence interval for a set of data can varysignificantly with the number of subsets used. The selection of anunsuitable number of subsets may cause incorrect results. Furthermore,this method requires significant storage capacity and computationalpower as the size of the data set increases. Finally, due to the natureof the computation, these intensive calculations have to be repeatedevery time additional data becomes available.

[0010] Many approaches have been developed to assist the selection ofthe number of subsets for the above batching method. However, they areusually very complicated and require a high level of expertise. Inaddition, the results of these approaches may differ from one another.Furthermore, the computational requirements increased ever further asthese approaches frequently require a significant statistical effort toanalyze the subsets and the relation therebetween.

[0011] A variant of the above conventional batching method, known asoverlapping batch means method, creates overlapping subsets. While thisvariant may have a slight improvement over the basic batching method, itstill requires a large data set, the selection of a number of subsets,significant storage and computational capacity. Furthermore, thecomplexity of the variant is still significant and requires significantstatistical knowledge.

BRIEF SUMMARY OF THE INVENTION

[0012] It is therefore an object of the present invention to permit theestimation of a measure of randomness of a function of at least onerepresentative value of at least one random variable, even for arelatively small size of data set to be used, in a reduced time.

[0013] The object may be achieved according to any one of the followingmodes of this invention. Each of these modes of the invention isnumbered like the appended claims, and depends from the other mode ormodes, where appropriate. This type of explanation about the presentinvention is for better understanding of some instances of a pluralityof technical features and a plurality of combinations thereof disclosedin this specification, and does not mean that the plurality of technicalfeatures and the plurality of combinations in this specification areinterpreted to encompass only the following modes of this invention:

[0014] (1) A method of estimating a measure of randomness of a functionof at least one representative value of at least one random variable,comprising:

[0015] a step of obtaining the at least one random variable;

[0016] a step of determining the at least one representative value ofthe obtained at least one random variable;

[0017] a step of determining a statistic of the obtained at least onerandom variable;

[0018] a step of determining a gradient of the function with respect tothe determined at least one representative value; and

[0019] a step of transforming the obtained statistic of the at least onerandom variable into a statistic of the function, using the determinedgradient.

[0020] As the result of the inventor's research, he has found that thereexists a statistical characteristic that, while a statistic of afunction of a random variable, which statistic may include a measure ofrandomness or dispersion, strongly tends to reflect a statistic of therandom variable, which statistic may include the measure of randomnessor dispersion, such that the statistic of the random variable isenlarged in the case of a steep gradient of the function of the randomvariable, the statistic of the function strongly tends to reflect astatistic of the random variable such that the statistic of the randomvariable is reduced in the case of a gentle gradient of the function ofthe random variable

[0021] In addition, the above research also revealed that, theutilization of the characteristic mentioned above would permit theestimation of a measure of randomness of a function of a representativevalue of a random variable, ensuring an accuracy thereof almost equal toone established in the use of the conventional batching methodaforementioned, with a smaller size of data used than in the batchingmethod, in a shorter time required than in the batching method.

[0022] On the basis of the above findings, in the above mode (1) of thepresent invention, at least one representative value of at least onerandom variable is determined and a statistic of the at least one randomvariable is determined. Furthermore, in the mode (1), a gradient of afunction of the at least one random variable with respect to thedetermined at least one representative value is determined, and, by theuse of the determined gradient, the determined statistic of the at leastone random variable is transformed into a statistic of the function.

[0023] Hence, the mode (1) would permit the estimation of a measure ofrandomness of a function of at least one representative value, by theuse of a smaller size of data used than in the conventional batchingmethod, in a shorter time required than in the batching method.

[0024] The term “representative value” may be defined, in the above mode(1) and other modes of the present invention, to mean a measure ofcentral tendency of a distribution of a plurality of individual datavalues belonging to the at least one random variable or the function,for instance.

[0025] Further, in the case of a plurality of random variables or aplurality of functions, the term “representative value” may be defined,in the above mode (1) and other modes of the present invention, to meana plurality of representative values for the plurality of randomvariables or functions, respectively, for instance.

[0026] In addition, the step of determining a gradient may beconstituted to exactly or approximately determine the gradient. Forexample, the step of determining a gradient may be adapted to determinea gradient of the function exactly at the at least one representativevalue, and may be adapted to determine a gradient of the function in thevicinity of the at least one representative value.

[0027] Furthermore, the term “function” is interpreted, in the abovemode (1) and other modes of the present invention, as an operator forassociating the at least one random variable with at least one othervariable, one example of which may be a performance function describedbelow, which function associates the at least one random variable with aperformance measure.

[0028] (2) The method according to the above mode (1), wherein the stepof transforming comprises transforming the statistic of the at least onerandom variable into the statistic of the function, such that thestatistic of the function responds to the statistic of the at least onerandom variable more sensitively in the case of the gradient being steepthan in the case of the gradient being gentle.

[0029] In the above mode (2), in light of the statistical characteristicaforementioned, which has been recognized by the inventor, a statisticof at least one random variable is transformed into a statistic of afunction, such that the statistic of the function responds to thestatistic of the at least one random variable more sensitively in thecase of the gradient being steep than in the case of the gradient beinggentle. (3) The method according to the above mode (1) or (2), whereineach one of the at least one representative value of the at least onerandom variable comprises at least one of an average, an arithmeticmean, a geometric mean, a median, a harmonic mean, and a mode, of eachone of the at least one random variable.

[0030] (4) The method according to any one of the above modes (1) to(3), wherein the step of determining the at least one representativevalue comprises determining the at least one representative value of theat least one random variable, upon truncating at least one part ofindividual data values belonging to the at least one random variable.

[0031] In the above mode (4), the at least one representative value isdetermined with the removal of abnormal data out of the plurality ofindividual data values by the application of truncation to the originalindividual data values, resulting in the improvement in an accuracy ofdetermining the at least one representative value, followed by theimprovement in an accuracy of estimating the randomness of the functionof the at least one random variable.

[0032] (5) The method according to any one of the above modes (1) to(4), wherein the statistic of each one of the at least one randomvariable comprises at least one of a standard deviation, a confidenceinterval, a set of data, a probability density function, and acumulative density function, of the each random variable.

[0033] (6) The method according to any one of the above modes (1) to(5), wherein the statistic of the function comprises at least one of astandard deviation, a confidence interval, a set of data, a probabilitydensity function, and a cumulative density function, of the function.

[0034] (7) The method according to any one of the above modes (1) to(6), further comprising a step of estimating the measure of randomnessof the function of the at least one representative value, on the basisof the statistic of the function.

[0035] (8) The method according to the above mode (7), wherein themeasure of randomness comprises a range of a confidence interval of thefunction of the at least one representative value.

[0036] (9) The method according to the above mode (7) or (8), applied toa simulation for discrete event, results of which simulation is used toeffect the method, wherein the step of estimating comprises estimatingthe measure of randomness using results of only one execution of thesimulation.

[0037] The above mode (9) would permit the estimation of the randomnessof the function of the at least one random variable in a shorter timethan estimated by the conventional batching method aforementioned.

[0038] Furthermore, this mode (9) would allow the reduction in timelength required for the randomness estimation described above for onesimulation, and as a result, this mode (9) would facilitate to performthe randomness estimation for other simulation within a given time.

[0039] Consequently, in the case where a plurality of simulations for asystem to be investigated on its performance is required for the aboverandomness estimation, this mode (9) would permit the randomnessestimation for those plurality of simulations in a shorter time than theconventional batching method mentioned before.

[0040] Thus, this mode (9) would also facilitate to compare theestimated measures of randomness for those plurality of simulationswithin a reduced time, facilitating an optimization of the system to beinvestigated using simulations, within a shorter time, at an improvedaccuracy.

[0041] (10) The method according to the above mode (9), wherein anaccuracy to be satisfied with the statistic of the function ispredetermined, and the step of determining a statistic comprises:

[0042] (a) determining the statistic of the at least one randomvariable, on the basis of a sum of individual data values belonging tothe at least one random variable;

[0043] (b) determining the statistic of the at least one random variableon the basis of the sum, upon adding to the sum at least one newindividual data value belonging to the at least one random variable;

[0044] (c) determining the statistic of the at least one random variablewhen at least one new individual data value belonging to the at leastone random variable becomes available during the simulation;

[0045] (d) transforming the determined statistic of the at least onerandom variable into the statistic of the function; and

[0046] (e) terminating the simulation when the predetermined accuracy issatisfied with the statistic of the function.

[0047] The above mode (10) would facilitate to monitor the increase inan accuracy of the statistic of the function of the at least one randomvariable as the simulation progresses.

[0048] In addition, this mode (10) would facilitate to automaticallyterminate the simulation when the predetermined accuracy of thestatistic of the function of the at least one random variable isreached.

[0049] (11) The method according to any one of the above modes (1) to(10), wherein the function is a function of a plurality of randomvariables, the step of transforming comprising:

[0050] (a) determining a measure of randomness of each one of the randomvariables at or in the vicinity of a representative value of each one ofthe obtained plurality of random variables, as the statistic of eachrandom variable;

[0051] (b) determining a measure of dependence between the plurality ofrandom variables; and

[0052] (c) transforming the determined measures of randomness of theplurality of random variables into a measure of randomness of thefunction, using the determined measure of dependence and the determinedgradient.

[0053] In the above mode (11), in the case of a plurality of randomvariables, the randomness of the function of the plurality of randomvariables is estimated by taking account of a measure of dependencebetween those random variables.

[0054] Subsequently, this mode (11) would allow, in the case of aplurality of random variables, the accurate estimation of the randomnessof the function of those random variables.

[0055] (12) The method according to the above mode (11), wherein themeasure of randomness of the each random variable comprises at least oneof a maximum likelihood estimator of a variance of the each randomvariable, an unbiased estimator of the variance, a maximum likelihoodestimator of a standard deviation of the each random variable, anunbiased estimator of the standard deviation, a variance of arepresentative value of the each random variable, a standard deviationof a representative value of the each random variable, a coefficient ofvariation of the each random variable, a general central moment of theeach random variable, a confidence interval of the each random variable,a set of data indicative of the each random variable, a probabilitydensity function of the each random variable, and a cumulative densityfunction of the each random variable.

[0056] (13) The method according to the above mode (11) or (12), whereinthe measure of dependence comprises at least one of an unbiasedestimator of a covariance of the plurality of random variables, amaximum likelihood estimator of the covariance, and a correlationcoefficient of the plurality of random variables.

[0057] (14) The method according to any one of the above modes (1) to(13), wherein the function is a function of a plurality of randomvariables, the step of transforming comprises transforming the obtainedstatistic of the plurality of random variables into the statistic of thefunction, without a calculation of a measure of dependence between theplurality of random variables.

[0058] In the above mode (14), in the case of a plurality of randomvariables, a statistic obtained for those random variables istransformed into a statistic of the function, without a calculation ofthe dependence between those random variables.

[0059] Thus, this mode (14) would permit, in the case where the numberof the at least one random variable is plural, and where the pluralityof random variables are independent of each other or are dependent fromeach other at a negligible low level, the estimation of the randomnessof the function of the random variables in a shorter time than when,upon the calculation of dependence between those random variables, thetransformation between statistics is performed.

[0060] (15) A method of determining a set of data of a function of arepresentative value of each one of at least one random variable, whichset of data allows an evaluation of a statistic of the function,comprising:

[0061] a step of obtaining a set of individual data values belonging toeach random variable, which set represents an approximation of adistribution of the each random variable;

[0062] a step of determining the representative value of the each randomvariable;

[0063] a step of determining a gradient of the function with respect tothe determined representative value; and

[0064] a step of transforming the obtained set of individual data valuesinto the set of data representing the function.

[0065] In the above mode (15), in light of the findings recognized bythe inventor of the present invention, as described with relation to theabove mode (1), a set of a plurality of individual data values belongingto the each random variable, which set represents an approximation of adistribution of the each random variable is obtained, and arepresentative value of the each random variable is determined.Furthermore, in this mode (15), a gradient of the function of the atleast one random variable with respect to the determined representativevalue is determined, and by the use of the determined gradient, theobtained set of individual data values for the at least one randomvariable is transformed into a set of data representing the values ofthe function.

[0066] Consequently, this mode (15) would permit the estimation of ameasure of randomness of a function of at least one random variable inthe form of a set of data representing the randomness, according tobasically the same principle as the one accepted in the above mode (1).

[0067] (16) The method according to the above mode (15), wherein thestep of transforming the set of individual data values of the eachrandom variable into the set of data representing the function, suchthat the set of data representing the function responds to the set ofindividual data values more sensitively in the case of the gradientbeing steep than in the case of the gradient being gentle.

[0068] (17) The method according to the above mode (15) or (16), furthercomprising a step of estimating a measure of randomness of the functionof the representative value, on the basis of the set of datarepresenting the function.

[0069] (18) The method according to the above mode (17), wherein themeasure of randomness comprises a range of a confidence interval of thefunction of the representative value.

[0070] (19) The method according to the above mode (17) or (18), appliedto a simulation for discrete event, results of which simulation is usedto effect the method, wherein the step of estimating comprisesestimating the measure of randomness using results of only one executionof the simulation.

[0071] The above mode (19) would provide basically the same operationand advantageous effects as the above mode (9) would.

[0072] (20) The method according to any one of the above modes (1) to(19), applied to an analysis of a plurality of business models to beaccepted in realizing a given business, wherein a function of at leastone of random variable is predetermined for each one of the plurality ofbusiness models, and the function of a representative value of the eachrandom variable for one of the plurality of business models is to becompared with the function of a representative value of the each randomvariable for another of the plurality of business models.

[0073] The above mode (20) would allow the determination of an accuracyof the function of the at least one random variable, for each businessmodel.

[0074] In addition, this mode (20) would permit the determination of thelikelihood of one business model outperforming another business model.

[0075] (21) A method of estimating a measure of randomness of at leastone random variable to satisfy a predetermined condition regarding ameasure of randomness of a function of at least one representative valueof the at least one random variable, the predetermined condition beingformulated to define a central location and a measure of dispersion, ofa distribution of the function, comprising:

[0076] a step of determining a gradient of the function with respect tothe defined central location; and

[0077] a step of determining the measure of randomness of the at leastone random variable, on the basis of the determined gradient and thedefined measure of dispersion.

[0078] As is apparent from the previous explanation regarding the abovemode (1), it is possible to mutually associate a measure of randomnessof at least one random variable, and a measure of randomness of afunction of at least one representative value of the at least one randomvariable. This means that, the use of a gradient of the function wouldpermit not only a forward estimation to estimate a measure of randomnessof the function of the at least one representative value of the at leastone random variable, from a measure of randomness of the at least onerandom variable, but also a backward estimation to estimate a measure ofrandomness of the at least one random variable, from a measure ofrandomness of the function of the at least one representative value ofthe at least one random variable.

[0079] In view of the above findings, in the above mode (21), acondition to be satisfied by a measure of randomness of a function of atleast one representative value of at least one random variable ispredetermined, where the predetermined condition defines a centrallocation of a distribution of the function, and a measure of dispersionof the distribution. Furthermore, in this mode (21), a gradient of thefunction with respect to the defined central location is determined, onthe basis of the determined gradient and the defined measure ofdispersion, and a measure of randomness of the at least one randomvariable.

[0080] (22) The method according to the above mode (21), wherein thestep of determining the measure comprises transforming the definedmeasure of dispersion into the measure of randomness of the at least onerandom variable, such that the measure of randomness of the at least onerandom variable responds to the defined measure of dispersion moresensitively in the case of the gradient being steep than in the case ofthe gradient being gentle.

[0081] In the above mode (22), by the use of a gradient of the function,according to a principle accompanied with necessary changes to oneaccepted in the above mode (2), the defined measure of dispersion of thefunction is transformed into a measure of randomness of the at least onerandom variable.

[0082] (23) The method according to the above mode (21) or (22), whereinthe measure of dispersion comprises at least one of a standarddeviation, a confidence interval, a set of data, a probability densityfunction, and a cumulative density function, of the function.

[0083] (24) The method according to any one of the above modes (21) to(23), wherein the measure of randomness of each one of the at least onerandom variable comprises at least one of a standard deviation, aconfidence interval, a set of data, a probability density function, anda cumulative density function, of the each random variable.

[0084] (25) A computer program to be executed by a computer to effectthe method according to any one of the above modes (1) to (24).

[0085] When a computer program according to the above mode (25) isexecuted by a computer, the same advantageous effects would be provided,according to basically the same principle as one accepted in a methodset forth in any one of the above modes (1) to (24).

[0086] The term “program” may be interpreted to include, not only a setof instructions to be executed by a computer so that the program mayfunction, but also any files and data to be processed by the computeraccording to the set of instructions.

[0087] (26) A computer-readable storage medium having stored therein thecomputer program according to the above mode (25).

[0088] When the program having been stored in a computer-readablestorage medium is executed by a computer, the same advantageous effectswould be provided, according to basically the same principle as oneaccepted in a method set forth in any one of the above modes (1) to(24).

[0089] The term “storage medium” may be realized in different types,including a magnetic recording medium such as a floppy-disc, an opticalrecording medium such as a CD and a CD-ROM, an optical-magneticrecording medium such as an MO, an unremovable storage such as a ROM,for example.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0090] The foregoing summary, as well as the following detaileddescription of preferred embodiments of the invention, will be betterunderstood when read in conjunction with the appended drawings. For thepurpose of illustrating the invention, there is shown in the drawingsembodiments which are presently preferred. It should be understood,however, that the invention is not limited to the precise arrangementsand instrumentalities shown. In the drawings:

[0091]FIG. 1 is graphs for explaining a relationship between a randomvariable X, a performance measure Z, a performance function f, and atangent function f′, which relationship is established in a method ofestimating a measure of randomness of a function of a random variableaccording to a first embodiment of the present invention;

[0092]FIG. 2 is a block diagram schematically illustrating a hardwarearrangement of a computer system used by a user thereof for effectingthe above method of FIG. 1;

[0093]FIG. 3 is a flow chart schematically illustrating arandom-variable-function-randomness estimation program executed by acomputer 20 of FIG. 2 to effect the above method of FIG. 1;

[0094]FIGS. 4A to 4G show equations (1) to (7), respectively, forexplaining the above estimation program of FIG. 3;

[0095]FIG. 5 is a histogram illustrating a frequency distribution of arandom variable in the above method of FIG. 1;

[0096]FIG. 6 is a graph schematically illustrating an event to which theabove method of FIG. 1 is applied;

[0097]FIG. 7 is a histogram illustrating a frequency distribution of arandom variable in the above method of FIG. 1, which random variable hasbeen truncated;

[0098]FIGS. 8A to 8F show equations (8) to (13), respectively, forexplaining the above estimation program of FIG. 3;

[0099]FIGS. 9A to 9C show equations (14) to (16), respectively, forexplaining the above estimation program of FIG. 3;

[0100]FIGS. 10A to 10D show equations (17) to (20), respectively, forexplaining the above estimation program of FIG. 3;

[0101]FIG. 11 is a flow chart schematically illustrating arandom-variable-function-randomness estimation program executed by acomputer to effect a method of estimating a measure of randomness of afunction of a random variable according to a second embodiment of thepresent invention;

[0102]FIG. 12 is a flow chart schematically illustrating arandom-variable-function-randomness estimation program executed by acomputer to effect a method of estimating a measure of randomness of afunction of a random variable according to a third embodiment of thepresent invention;

[0103]FIGS. 13A to 13C show equations (21) to (23), respectively, forexplaining the above estimation program of FIG. 12; and

[0104]FIG. 14 is a flow chart schematically illustrating arandom-variable-randomness estimation program executed by a computer toeffect a method of estimating a measure of randomness of a randomvariable according to a fourth embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

[0105] Several presently preferred embodiments of the invention will bedescribed in detail by reference to the drawings in which like numeralsare used to indicate like elements throughout.

[0106] [First Embodiment]

[0107] Referring first to FIG. 1, there will be described atechnological principle accepted in a first embodiment of the presentinvention in the form of a method of estimating a measure of randomnessof a function of at least one random variable (hereinafter referred tosimply as “randomness estimating method”) by a graph thereon. In thefirst embodiment, a measure of randomness of a function of at least onerepresentative value of at least one random variable is estimated usinga gradient of the function, and thereby transforming a measure ofrandomness of the at least one random variable into a measure ofrandomness of the function of the at least one representative value ofthe at least one random variable.

[0108] Described in more detail, in the first embodiment, therepresentative value of each random variable is used in the form of amean thereof, and the function of the mean is used in the form of aperformance function which is a function of a mean time betweenoccurrences and which is to derive a frequency of occurrence from themean time between under the function. In FIG. 1, a two-dimensionalcoordinate system is shown with a random variable x being taken alongthe horizontal axis, and with a performance measure z of a system to beinvestigated by a simulation, being taken along the vertical axis. Inthe coordinate system, the performance function in the form of z=f(X) isshown by a broken line, while a tangent function in the form of z=f′(X)representing a tangent (i.e., a tangent line or plane) of theperformance function is shown by a solid line.

[0109] Referring next to FIG. 2, where is schematically illustrated by ablock diagram a hardware arrangement of a computer system 10 to be usedby a user to effect the aforementioned randomness estimating methodaccording to the first embodiment.

[0110] The computer system 10 is constructed, as well known to thoseskilled in the art, to have a computer 20 so configured that aprocessing unit (referred to as “PU” in the drawings and thedescription) 12 and a storage 14 are connected with each other with abus 16. The computer 20 is connected with an input device 30 equippedwith a pointing device in the form of a mouse and a keyboard, and anoutput device 40 displaying an image on a screen thereof The storage 14is constructed to contain a recording medium such as a ROM, a RAM, amagnetic disc, an optical disc, etc. The user of the computer system 10inputs data to be required, into the computer 20 through the inputdevice 30. In response to the input operation, results of dataprocessing performed by the computer 20 are visualized to present beforethe user by means of the output device 40.

[0111] The storage 14 has already stored therein arandom-variable-function estimation program (referred to as “estimationprogram”) to be executed by the PU 12 in order to effect the randomnessestimation method according to the first embodiment of the presentinvention. The storage 14 is designed to store therein data to be usedduring the execution of the estimation program by the PU 12, whereappropriate.

[0112] Referring next to FIG. 3, there is schematically illustrated by aflow chart the estimation program mentioned above. While this programwill be described below in reference to the flow chart, definitions ofseveral symbols used in the estimation program will be first explained.

[0113] X: random variable as constructed by a set of individual datavalues x_(i)

[0114] x_(i): each one of the individual data values belonging to randomvariable X

[0115] Y: random variable as constructed by a set of individual datavalues y_(i)

[0116] y_(i): each one of the individual data values belonging to randomvariable Y

[0117] i: numeral of each individual data value

[0118] n: set size, namely, the number of individual data values in eachset thereof

[0119] Z: performance measure of a system to be investigated by asimulation

[0120] a: confidence level

[0121] Z_(n−1),(1−a)/2: chi-square distribution (1−a)/2 quantile for setsize n

[0122] E[X]: mean (or expected value) of random variable X

[0123] E[Y]: mean (or expected value) of random variable Y

[0124] E[Z]: mean (or expected value) of performance function orperformance measure

[0125] S[X]: standard deviation of random variable X

[0126] S[Y]: standard deviation of random variable Y

[0127] S[Z]: standard deviation of performance function

[0128] f(E[X]): general function of mean (or expected value) of randomvariable X

[0129] f(E[X],E[Y], . . . ): general function of means (or expectedvalues) of random variables X, Y, . . .

[0130] CI[X]: confidence interval half width of random variable X

[0131] Cov[X, Y]: covariance of random variables X, Y

[0132] Corr[X, Y]: correlation coefficient of random variables X, Y

[0133] W: frequency of occurrence

[0134] P: duration divided by time between start of durations' cycles

[0135] E[W]: mean (or expected value) of frequency W as a function ofthe mean of random variable X

[0136] E[P]: mean (or expected value) of percentage P as a function ofthe means of random variables X and Y

[0137] The randomness estimating method according to the firstembodiment of the present invention requires at least one randomvariable, in which a plurality of individual data values thereof arerandomly distributed. Equation (1) of FIG. 4A represents, by way of anexample where the random variable is X, that the random variable X isconstructed to form a set of individual data values x_(i). FIG. 5 showsa histogram of an example of a set of individual data values in a randomlog-normal distribution manner, for the better understanding of arelationship (i.e., a frequency distribution) between a random variableand a frequency thereof.

[0138] The randomness estimating method according to the firstembodiment also requires a performance function of the means of at leastone random variable. Equation (2) of FIG. 4B shows such a performancefunction as a general function f, where a performance measure Z is afunction of random variables X, Y, . . . A frequently used embodiment ofthe performance measure is for example a frequency of occurrence, wherethe frequency W is a function of the mean E[X] of a randomly distributedtime X between occurrences as shown by equation (3) in FIG. 4C. Anotherfrequently used embodiment of the performance measure is for example apercentage, where the percentage P is a function of the mean E[X] of arandomly distributed time X between occurrences, and the mean E[Y] of arandomly distributed duration Y of the occurrences as shown by equation(4) of FIG. 4D.

[0139] Referring next to FIG. 6, there are schematically illustrated byway of example several individual data values x_(i) belonging to timebetween occurrences X, which serves as a first random variable, andseveral individual data values y_(i) belonging to duration ofoccurrences Y, which serves as a second random variable.

[0140] The randomness estimation program will be described by referringto the flow chart of FIG. 3.

[0141] This program is initiated with step S1 where informationregarding a random variables, which is to say, a set of datarepresenting individual data values x_(i), y_(i) of the random variablesis prepared.

[0142] It is to be added that, if necessary, the data set of the randomvariables may be truncated by removing a percentage of the smallestand/or largest individual data values. This is schematically shown inFIG. 7, where an unequal part of the smallest and the largest individualdata values of the example in FIG. 5 has been removed.

[0143] It is to be also added that, for example, a set of data of therandom variables may be removed during a warming up period (i.e.,considered as a transition period of the aforementioned system inoperation).

[0144] Step S1 described above is followed by step S2 to calculate a sumof all individual data values x_(i), y_(i) belonging to each randomvariable.

[0145] Then, in step S2, the calculated sum is divided by the number nof the individual data values x_(i), y_(i), to thereby determine thearithmetic mean of all the individual data values as the mean E[X],E[Y].

[0146] Subsequently, step S3 is implemented to determine a standarddeviation S[X] of all the individual data values belonging to randomvariable X, and a standard deviation S[Y] of all the individual datavalues belonging to random variable Y. There are different estimators ofa standard deviation, for example, an unbiased estimator of the standarddeviation, or a maximum likelihood estimator of the standard deviation.While the unbiased estimator of the standard deviation is preferred,either estimator can be used. Similar is true for a variance, which, bydefinition, is merely a square of a standard deviation. For independentand identically distributed data, the unbiased estimator of the standarddeviation S[X] can be calculated as shown by equation (6) in FIG. 4F.The maximum likelihood estimator SML[X] can be calculated as shown byequation (7) in FIG. 4G.

[0147] Then, in step S4, a variable analysis is performed to determine ameasure of dependence between the random variables, in the case of aplurality of random variables. If the random variables are independent,no measure of dependence therebetween is needed. However, if the randomvariables are not independent, a measure of dependence therebetween isnecessary. A covariance thereof is used as a measure of dependencetherebetween. A biased estimator of the covariance of two randomvariables is calculated as shown by equation (8) in FIG. 8A.

[0148] It is to be added that, in step S4, instead of the unbiasedestimator of the covariance Cov[X, Y], a maximum likelihood estimator ofthe covariance COV_(ML)[X, Y] can be used, as shown by equation (9) inFIG. 8B. The difference therebetween is small, and either estimator canbe used in the randomness estimating method according to the firstembodiment of the present invention.

[0149] It is to be also added that, in step S4, alternatively, acorrelation coefficient Corr[X, Y] can be substituted for the covarianceCov[X, Y] as shown by equation (10) in FIG. 8C.

[0150] Step S4 described above is followed by step S5 to determine agradient or a slope of the performance function mentioned above of themeans of all the individual data values for all the random variables X,Y, at the mean values E[X], E[Y]. These gradients can be visualized astangents at the points of the mean values. The example is shown in FIG.1, where a frequency performance function (indicated by a broken line)is plotted with the tangent (indicated by a solid line) at the mean timebetween occurrences E[X]. FIG. 1 further shows a frequency distributionof the random variable X by a solid line convex upward, a frequencydistribution of the performance measure Z by a solid line convexrightward, respectively.

[0151] The slope for one random variable is determined bydifferentiating the performance function with respect to this randomvariable. This has to be done for all the random variables, giving oneslope for each random variable.

[0152] It is to be added that equation (3) in FIG. 4C shows, as oneexample, a performance function g for a frequency W, while equation (11)in FIG. 8D shows a differential of the performance function g withrespect to the single random variable X. Equation (4) in FIG. 4D shows,as another example, a performance function h for a percentage P, wherethe percentage P is calculated on the basis of two random variables X,Y. Two differentials of the performance function h are shown byequations (12) and (13) in FIGS. 8E, 8F, respectively.

[0153] However, it is not essential in using the present invention toexactly determine the gradient of the performance function so as to beequal to the gradient of the performance function exactly at the meanvalue. The present invention may be used to approximately determine thegradient of the performance function so as to be equal to the gradientof the performance function at one of the plurality of individual datavalues close to the mean thereof. Any approach to determine the gradientwould not limit the scope of the present invention.

[0154] Step S5 previously described is followed by step S6 to determinethe mean value E[Z] of the performance function. This can be done easilyby applying the corresponding performance function g, h to the meanE[X], E[Y] of the corresponding random variable, as shown for an exampleof the frequency W by equation (3) in FIG. 4C, and for an example of thepercentage P by equation (4) in FIG. 4D.

[0155] Step S7 is subsequently executed to translate or transform thestandard deviation S[X], S[Y] of the random variables X, Y as calculatedin step S3 into the standard deviation S[Z] of the performance function,using the gradient determined in step S5.

[0156] For one random variable, this can be done easily as shown byequation (14) in FIG. 9A. Equation (14) indicates that an operation tomultiply the standard deviation S[X] by the differential df/dE[X]permits the standard deviation S[X] to be translated into the standarddeviation S[Z] of the performance measure Z. It is to be noted that thesquare root of the square ensures a positive standard deviation.

[0157] For two random variables, equation (15) shown in FIG. 9B can beused. Equation (15) indicates that an operation to multiply the standarddeviation S[X] by the differential df/dE[X], an operation to multiplythe standard deviation S[Y] by the differential df/dE[Y], and anoperation to multiply the covariance Cov[X, Y] by the differentialsdf/dE[X] and df/dE[Y] corporate to permit two standard deviations S[X]and S[Y] to be translated into one standard deviation S[Z] of theperformance measure Z. Equation (15) has a term of the covariance,namely, Cov[X, Y], a value of which is used as one calculated in stepS4.

[0158] It is to be noted that if those random variables are independent,the covariance Cov[X, Y] is zero and the term can be dropped. However,irrespective of whether those random variables are independent or not,the present invention may be practiced such that the term of thecovariance is neglected in equation (15).

[0159] For more than two random variables, more complex statisticalmethods of the random variables have to be used.

[0160] Summing up, not the performance function is used to reflect thestandard deviation of the random variables on the standard deviation ofthe performance function, but instead the slope is used to reflect thestandard deviation of the random variables on the standard deviation ofthe performance function.

[0161] Described more specifically, in the present embodiment, thestandard deviation of the random variables is transformed into thestandard deviation of the performance function according to a ratioresponsive to the gradient of the performance function, by takingaccount of a statistical characteristic that, while a standard deviationof a random variable strongly tends to be transformed into a standarddeviation of a performance function in an enlarged manner when agradient of the performance function is steep, a standard deviation of arandom variable strongly tends to be transformed into a standarddeviation of a performance function in a reduced manner when a gradientof the performance function is gentle.

[0162] Step S7 is followed by step S8 to normalize the standarddeviation of the performance measure of the performance function for thenumber n of the individual data values of the set thereof for the randomvariables. The current standard deviation S[Z] is a theoretical value ofeach individual data value. Subsequently, to obtain the standarddeviation of the mean value of the performance function, the standarddeviation of the performance function has to be divided by the squareroot of the number n of individual data values as shown by equation (16)in FIG. 9C, resulting in a normalized standard deviation S_(mean)[Z].This value is automatically included in the calculation of a confidenceinterval in the next step S9.

[0163] Step S9 uses the mean E[Z] of the performance function determinedin step S6, and the standard deviation of the performance functiondetermined in step S8, in order to determine a confidence interval ofthe performance measure of the performance function. This confidenceinterval depends on the selected confidence level a, the size of thesamples n (i.e., the set size), the standard deviation of the generalperformance function S[Z], and the mean of the general performancefunction E[Z]. A confidence interval half width CI[Z] is calculated asshown by equation (17) in FIG. 10A, where Z_(n−1), (1−a)/2 is the(1−a)/2 Quantile of the chi-square distribution for n−1 elements. Theconfidence interval half width CI[Z] gives a confidence interval rangeas shown by equation (20) in FIG. 10D.

[0164] It is to be added that there are additional approaches tocalculate a confidence interval, as for example the use of a student tdistribution. Any approach to calculate a confidence interval would notlimit the scope of the present invention.

[0165] Then, one cycle of execution of this program is terminated.

[0166] The present embodiment will be described more specifically, byreferring to one example to which the present embodiment is practicallyapplied, comparing with the aforementioned batching method serving asone of conventional methods for the same purposes.

[0167] The applied example of the present embodiment assumes thesimulation of a manufacturing system. Within this sample, the failuresof a certain machine are analyzed. To simplify the problem, the machineis supposed to be placed either in a state of repair following afailure, or in a state of being available. A simulation is performed todetermine the performance of the simulated manufacturing system.

[0168] Within this simulation, the time between the occurrence of afailure x is recorded, creating a set of data of times between failuresX. The time to repair a failure is also recorded, creating a set of dataY. It is to be noted that the repair time y is part of the time betweenfailures x, as shown in FIG. 6.

[0169] A failure is a rare event, with a long average time between theoccurrences of two failures. Subsequently, even after running thesimulation for a long time, only a few failures occurred. For theselected long simulation time, a set of 16 independent and identicallydistributed individual data values for random variables X and Y wereobtained after removing the warming up period. This relates to step S1in the flow chart of FIG. 3.

[0170] A small set of individual data values fails to allow the use ofthe above-mentioned batching method. While it is possible to divide the16 values into 4 batches of 4 values, the resulting batch means would behighly inaccurate. The standard deviation of the batch means woulddepend on only 4 batch means, and therefore the standard deviation wouldalso be highly inaccurate. Subsequently, it would be necessary to run alonger simulation to permit the batch means method. Since the simulationto obtain 16 values was already very long, a longer simulation isundesirable.

[0171] However, the present embodiment previously described allows thecalculation of a confidence interval without further effort.

[0172] In the present embodiment, an initial analysis by the abovesimulation of two random variables X and Y using equations (5) and (6)in FIGS. 4E, 4F and equations (8) and (10) in FIGS. 8A, 8C gives thefollowing values for the arithmetic means E[X], E[Y], the unbiasedstandard deviations S[X], S[Y], the unbiased covariance Cov [X, Y], andthe unbiased correlation coefficient Corr[X, Y]:

[0173] E[X]=109 min

[0174] E[Y]=15 min

[0175] S[X]=53 min

[0176] S[Y]=10 min

[0177] Cov[X, Y]=105 min

[0178] Corr[X, Y]=0.1981

[0179] The correlation coefficient Corr[X, Y] indicates a positivedependence of the random variables X and Y, that is, a relatively longtime between failures is likely to be followed by a relatively long timeto repair the machine. This relates to steps S2, S3, and S4 in the flowchart in FIG. 3.

[0180] In step S5, the gradient of the performance functions for X and Yis calculated. Based on the mean value of the time between failures X, afailure frequency W is calculated according to equation (3) in FIG. 4C.Also, a percentage P of the time during which the machine is underrepair is calculated as a function of the means as shown as equation (4)in FIG. 4D. The derivatives of the performance functions are shown asequations (11), (12) and (13) in FIGS. 8D, 8E, 8F, respectively. Thevalues of the derivatives at the mean are shown below:

[0181] dW/dE[X]=0.00008417

[0182] dP/dE[X]=0.001263

[0183] dP/dE[Y]=0.009174

[0184] In step S6, the mean values of the performance functions aredetermined. Accordingly, the value of the frequency W is 0.009174failures per minute and the value of the percentage P is 13.76%.

[0185] In step S7, the standard deviations of the performance value ofthe performance functions are calculated according to equation (14) inFIG. 9A for the frequency W, equation (15) in FIG. 9B for the percentageP. Subsequently, the standard deviation of the frequency W is 0.004461failures per minute and the standard deviation of the percentage P is10.23%.

[0186] However, these standard deviations are with respect to the size nof the data set. Then, in step S8, the standard deviations of the meansare calculated according to equation (16) in FIG. 9C for thenormalization. Subsequently, the standard deviation of the mean of thefrequency W is 0.001115 and the standard deviation of the mean of thepercentage P is 2.557%.

[0187] Finally, in step S9, a confidence interval is constructedaccording to equation (17) in FIG. 10A. A confidence level a of 95% wasselected. The confidence interval half width for the frequency W is0.002377 and the confidence interval half width for the percentage P is5.450%. This relates to a confidence interval of the frequency infailures per minute as shown below. This confidence interval is alsogiven using a unit of the number of failures per eight hour shift.

[0188] 0.004461∓0.002377 Failures/Minute

[0189] 2.14∓1.14 Failures/Shift

[0190] Similarly, the confidence interval of the percentage P can begiven as shown below:

[0191] 10.2%∓5.45%

[0192] Therefore, the present embodiment allowed the calculation of aconfidence interval for the set of data, even though a standard batchingmethod cannot be used due to the small sample size. The resultingconfidence interval can be used to decide if the information is accurateenough or if additional simulation is necessary to collect more data andto improve the accuracy of the results.

[0193] It will be understood from the above explanation that step S1provides an example of the step of obtaining set forth in the above mode(1), step S2 provides an example of the step of determining the at leastone representative value set forth in the same mode, step S3 provides anexample of the step of determining a statistic set forth in the samemode, step S5 provides an example of the step of determining a gradientset forth in the same mode, and steps S4, S7, and S9 corporate toprovide an example of the step of transforming set forth in the samemode.

[0194] [Second Embodiment]

[0195] There will next be described a method of estimating a measure ofrandomness of a function of at least one representative value of atleast one random variable, constructed according to a second embodimentof this invention. However, since the second embodiment is similar tothe first embodiment in a hardware construction for the estimation ofthe measure of randomness except a software construction for theestimation of the measure of randomness, especially arandom-variable-function-randomness estimation program to be executed bya computer corresponding to the computer 20 in the first embodiment,only this program will be described in detail.

[0196] In the first embodiment, a standard deviation of at least onerandom variable is used to determine a measure of randomness of the atleast one random variable. Described more specifically, a standarddeviation as one type of statistic representing a measure of randomnessof the at least one random variable, and afterward, the determinedstandard deviation is converted into a standard deviation as one type ofstatistic representing a measure of randomness of a performancefunction. Finally, a confidence interval of the performance function isdetermined.

[0197] In the present embodiment, a confidence interval half width ofthe at least one random variable is used to determine a measure ofrandomness of the at least one random variable. In other words, aconfidence interval half width as one type of statistic representing ameasure of randomness of the at least one random variable is determined,and subsequently, the determined confidence interval half width istransformed into a confidence interval half width as one type ofstatistic representing a measure of randomness of the performancefunction.

[0198] Referring next to FIG. 11, there is schematically illustrated bya flow chart the random-variable-function-randomness estimation programmentioned above in the present embodiment.

[0199] Steps S31, S32, and S33 of this program are executed in such amanner as in steps S1, S2, and S3 of the corresponding program in thefirst embodiment.

[0200] This program then proceeds to step S34 in which a confidenceinterval half width is calculated for all the random variables X, Yobtained in step S31. A standard equation as shown as equation (17) inFIG. 10A can be used, using the standard deviation S[X], S[Y] of therandom variables X, Y instead of the standard deviation S[Z] of theperformance measure Z of the performance function. The confidenceinterval half width is also a measure of variation of the randomvariables X, Y. A confidence level a has to be chosen to calculate aconfidence interval half width.

[0201] Subsequently, in step S35, the covariance Cov[X, Y] of the randomvariables X, Y is calculated in such a manner as in step S4 of the firstembodiment.

[0202] Then, in step S36, the correlation coefficient Corr[X, Y] iscalculated if more than one random variable is used in the performancefunction. Here, the performance function is assumed to be a function oftwo random variables X, Y. The calculation of the correlationcoefficient Corr[X, Y] is based on the covariance Cov[X, Y] asdetermined in step S35, and the standard deviations S[X], S[Y] of therandom variables X, Y as determined in step S33. Equation (10) in FIG.8C shows the functional relation between the correlation coefficientCorr[X, Y], the covariance Cov[X, Y], and the standard deviations S[X],S[Y].

[0203] Afterward, in step S37, gradients of the performance function ofthe means E[X], E[Y] are determined for all the random variables X, Y,at the means E[X], E[Y] determined in step S32, as in such a manner asin step S5 of the first embodiment.

[0204] Step S37 is followed by step S38 to determine the mean value E[Z]of the performance function in such a manner as in step S6 of the firstembodiment.

[0205] Subsequently, in step S39, by the use of the gradients determinedin step S37, the confidence interval half width CI of the randomvariables X, Y directly (i.e., not by way of a standard deviation as onetype of statistic) into a confidence interval half width CI of theperformance function.

[0206] Described more specifically, for one random variable, the abovetranslation can be done easily as shown by equation (18) in FIG. 10B. Itis to be note that the square root of the square ensures a positiveconfidence interval half width CI.

[0207] For two random variables, the calculation of the confidenceinterval half width CI is calculated using equation (19) in FIG. 10C.Equation (19) has a term of the Corr[X, Y]. It is to be noted that ifthe random variables are independent, the correlation coefficientCorr[X, Y] is zero and the term can be dropped.

[0208] Whether the number is the random variable(s) is one or two, thegradient is used in calculating the confidence interval half width CI,as shown by equations (18) and (19) in FIGS. 10B, 10C.

[0209] If more than two random variables are used, more complexstatistical approaches have to be used.

[0210] Subsequently, in step S40, the standard deviation S[Z] of theperformance function of the mean in such a manner as in step S8 of thefirst embodiment.

[0211] Following step S40, in step S41, the confidence interval of theperformance function is calculated on the basis of the confidenceinterval half width CI determined in step S39, and the mean value E[Z]determined in step S38. The function to calculate the confidenceinterval is shown as equation (20) in FIG. 10D.

[0212] Then, one cycle of the execution of this program is terminated.

[0213] It will be understood from the above explanation that step S31provides an example of the step of obtaining set forth in the above mode(1), step S32 provides an example of the step of determining the atleast one representative value set forth in the same mode, steps S33 andS34 corporate to provide an example of the step of determining astatistic set forth in the same mode, step S37 provides an example ofthe step of determining a gradient set forth in the same mode, and stepsS35, S36, S39, and S41 corporate to provide an example of the step oftransforming set forth in the same mode.

[0214] [Third Embodiment]

[0215] There will next be described a method of estimating a measure ofrandomness of a function of at least one representative value of atleast one random variable, constructed according to a third embodimentof this invention. However, since the third embodiment is similar to thefirst and second embodiments in a hardware construction for theestimation of the measure of randomness except a software constructionfor the estimation of the measure of randomness, especially arandom-variable-function-randomness estimation program to be executed bya computer corresponding to the computer 20 in the first embodiment,only this program will be described in detail.

[0216] The first and second embodiments of the present invention requirea standard deviation of at least one random variable, for the estimationof a measure of randomness of a performance function of the at least onerandom variable.

[0217] On the contrary, the presented third embodiment does not use astandard deviation of at least one random variable at all, in order toestimate a measure of randomness of a performance function of the atleast one random variable. Rather, the present embodiment creates atangential equation to the mean of the performance function, andtranslates a set of a plurality of individual data values belonging toeach random variable, into a set of individual data values representingthe performance function, using the created tangential equation.

[0218] Summing up, the present embodiment determines a set of aplurality of individual data values of each random variable, as astatistic representing a measure of randomness of the performancefunction of the random variables.

[0219] Referring next to FIG. 12, there is schematically illustrated bya flow chart the random-variable-function-randomness estimation programmentioned above in the present embodiment.

[0220] Steps S51, S52, and S53 of this program are executed in such amanner as in steps S1, S2, and S5 of the first embodiment of the presentinvention.

[0221] These steps of this program is followed by step S54 in which atangential equation is determined on the basis of the means of therandom variables, and the gradient of the performance function.

[0222] The number of dimensions of the tangential equation equals thenumber of the random variables. Described more specifically, if onerandom variable is used, the tangential equation defines a line. If tworandom variables are used, the tangential equation defines a plane.Similar tangents can be constructed for higher order dimensions of thetangential equation, although the visualization is difficult. A generaltangential equation f′ for one random variable is shown as equation (21)in FIG. 13A, while a general tangential equation f′ is shown as equation(22) in FIG. 13B.

[0223] Subsequently, in step S55, the sets of individual data values forall the random variables are entered in the tangential equationdetermined in step S54. The entry produces a set of data values Z′ forthe performance function of the mean values, on the basis of theindividual data values x, y of the random variables X, Y The tangentialequation is shown as equation (23) in FIG. 13C.

[0224] Step S55 is followed by step S56 in which a standard deviation iscalculated for the set of data values Z′ of the performance function. Bydefinition, the mean of the set of data values Z′ equals the performancefunction of the mean values of the random variables. The standarddeviation is calculated using equation (6) or (7) in FIG. 4F or 4G forthe unbiased or the maximum likelihood estimator.

[0225] Afterward, in step S57, a confidence interval is calculated onthe basis of the standard deviation of the set of data values Z′, andthe number of individual data values in the data set Z′. The calculationis done using a standard equation as shown as equation (17) in FIG. 10Ato calculate the confidence interval half width, and equation (20) inFIG. 10D to calculate the confidence interval.

[0226] It is important to note that, in this embodiment, a complete setof individual data values is available for the performance function ofthe means, allowing the use of more complex and sophisticated approachto determine the confidence interval. For example, it is possible totake the shape of the distribution of the individual data values intoaccount to obtain different confidence interval half widths for regionsabove and below the mean of the performance function.

[0227] It will be understood from the above explanation that step S51provides an example of the step of obtaining set forth in the above mode(1), and an example of the step of determining a statistic set forth inthe same mode, step S52 provides an example of the step of determiningthe at least one representative value set forth in the same mode, stepS53 provides an example of the step of determining a gradient set forthin the same mode, and steps S54 to S57 corporate to provide an exampleof the step of transforming set forth in the same mode.

[0228] It will be also understood from the above explanation that stepS51 provides an example of the step of obtaining set forth in the abovemode (15), step S52 provides an example of the step of determining therepresentative value set forth in the same mode, step S53 provides anexample of the step of determining a gradient set forth in the samemode, and steps S54 and S55 corporate to provide an example of the stepof transforming set forth in the same mode.

[0229] While the present invention has been described in detail in itspresently preferred embodiments, these embodiments would provide thefollowing advantageous results optionally or collectively:

[0230] (a) these embodiments allow the calculation of a measure ofrandomness of the function of means of the random variables even fordata sets of a relatively small size (e.g., the minimum size is two andthe recommended size is at least five);

[0231] (b) these embodiments allow the calculation of a measure ofrandomness of the function of the means with greatly improved accuracyover the conventional batching method previously mentioned;

[0232] (c) these embodiments allow the calculation of a measure ofrandomness of the function of the means with less effort than theconventional batching method;

[0233] (d) these embodiments calculate an updated measure of randomnessof the function of the means with minimal effort if new data becomesavailable;

[0234] (e) these embodiments allow the calculation of a measure ofrandomness of the function of the means with a minimal need for storageand computational power;

[0235] (f) the ease of implementing the preferred embodiments mentionedabove into an automated software program when compared with the previousbatching method, allowing for a simple and reliable calculation of thevalidity of the results of a set of data obtained from a softwaresimulation, for example;

[0236] (g) with respect to discrete event simulation, a singlesimulation will suffice to calculate a confidence interval, whereas theprevious batching method requires multiple simulations or a longsimulation split into multiple batches, thus saving simulation time andallowing the comparison of more simulations within a given time;

[0237] (h) with respect to discrete event simulation, a confidenceinterval can be calculated even if there exists only a small number ofindividual data values, for example, in the case of rare events, thevalidity of the results of computational analysis can be determined fora shorter simulation; and

[0238] (i) with respect to discrete event simulation, due to the smallcomputation and storage requirements mentioned above, it is possible tocalculate a confidence interval which updates as the simulationprogresses, i.e., it is possible to monitor the reduction of theconfidence interval width as the simulation progresses, where theinformation can be used for example to determine when a certain requiredaccuracy is reached and the simulation can be stopped.

[0239] It is to be added that the aforementioned embodiments of thepresent invention may be practiced in such form to calculate theconfidence interval, and to use the calculated confidence interval forautomatically updating the confidence interval during the progress ofthe simulation. In the form, it is possible to calculate the mean, thestandard deviation, and the correlation on the basis of sums ofindividual data values. Subsequently, if additional individual databecomes available, those sums have to be merely updated to create a newvalue of the confidence interval. Therefore, it is possible to calculatethe confidence intervals as the simulation progresses, with little or noeffort.

[0240] The above form of the preferred embodiments also allows anautomated simulation termination according to a required confidencelevel. During the creation of the simulation model, the desiredconfidence interval half widths of one or more simulation parameters arespecified. During the simulation, the confidence interval half widths ofthese simulation parameters are continuously updated. If the actual orupdated confidence interval half widths are equal to or less than thedesired confidence interval half widths for all the simulationparameters, the simulation is terminated.

[0241] The above form of the preferred embodiments would solve a bigproblem of current simulation methodologies, where the exact time lengthof a simulation is difficult to determine. This form would allow anautomatic termination of the simulation upon reaching a predefinedaccuracy criterion on results of the simulation.

[0242] It is to be also added that, in general, to calculate aconfidence interval using the conventional batching method requires atleast 5, generally 10 to 30 simulations. The preferred embodimentsmentioned above allow the calculation of a confidence interval for afunction of at least one mean, using only one simulation.

[0243] It is to be further added that, the preferred embodimentsmentioned above allow the calculation of confidence intervals for asmall set of individual data values, for example, a set of only 10individual data values. The conventional batching method cannotcalculate a remotely valid confidence interval for such a small set ofindividual data values.

[0244] It is to be still further added that, the preferred embodimentsmentioned above allow the calculation of a confidence interval withalmost the same range with the case where the conventional batchingmethod calculates the confidence interval on the basis of a large set ofindividual data values during multiple simulations, although thepreferred embodiments calculate the confidence interval on the basis ofa small set of individual data values during one simulation.

[0245] There may exist the case where information regarding adistribution of random variables is not available as a set of data, butrather available as a probability density function. In this case, thepreferred embodiment of the present invention described above would haveto be modified to determine the mean and deviation based on theprobability density function.

[0246] [Fourth Embodiment]

[0247] There will next be described a method of estimating a measure ofrandomness of at least one random variable to satisfy a predeterminedcondition regarding a measure of randomness of a function of arepresentative value of the at least one random variable, constructedaccording to a fourth embodiment of this invention. However, since thefourth embodiment is similar to the previous three embodiments in ahardware construction for the estimation of the measure of randomnessexcept a software construction for the estimation of the measure ofrandomness, especially a random-variable-randomness estimation programto be executed by a computer corresponding to the computer 20 in thefirst embodiment, only this program will be described in detail.

[0248] It is possible to mutually associate a measure of randomness ofat least one random variable, and a measure of randomness of a functionof a representative value of the at least one random variable. Thismeans that, the use of a gradient of the function would permit not onlya forward estimation to estimate a measure of randomness of the functionof the representative value of the at least one random variable, from ameasure of randomness of the at least one random variable, but also abackward estimation to estimate a measure of randomness of the at leastone random variable, from a measure of randomness of the function of therepresentative value of the at least one random variable.

[0249] In addition, it may be of use not to calculate a measure ofrandomness of a function of a mean based on random variables, but toreverse the equations described above and to obtain desired statisticalproperties of random variables which would be needed to obtain a certainmeasure of randomness of the function.

[0250] While the previous three embodiments of the present inventionperform the forward estimation mentioned above, the presented fourthembodiment performs the backward estimation also mentioned above.

[0251] In the fourth embodiment, the predetermined condition withrespect to the randomness of the performance function has beenformulated to define a central location and a measure of dispersion, ofa distribution of the performance function.

[0252] Referring next to FIG. 14, there is schematically illustrated theaforementioned random-variable-randomness estimation program in thepresent embodiment.

[0253] This program is initiated with step S71 in which data for thepredetermined condition is read from the storage 14. The data has beenstored therein.

[0254] In step S72, a gradient of the performance function with respectto the defined central location is determined by using the equationshown as for example equation (2), (3) or (4) in FIG. 4.

[0255] In step S73, a confidence interval as a measure of randomness ofthe at least one random variable is determined on the basis of thedetermined gradient and the defined measure of dispersion. Thedetermination is to transform the defined measure of dispersion into theconfidence interval of the at least one random variable, such that theconfidence interval responds to the defined measure of dispersion moresensitively in the case of the gradient being steep than in the case ofthe gradient being gentle.

[0256] Then, one cycle of execution of this program is terminated.

[0257] It will be understood from the above explanation that step S72provides an example of the step of determining a gradient set forth inthe above mode (21), and step S73 provides an example of the step ofdetermining the measure of randomness set forth in the same mode.

[0258] It will be appreciated by those skilled in the art that changescould be made to the embodiments described above without departing fromthe broad inventive concept thereof. It is understood, therefore, thatthis invention is not limited to the particular embodiments disclosed,but it is intended to cover modifications within the spirit and scope ofthe present invention as defined by the appended claims.

What is claimed is:
 1. A method of estimating a measure of randomness ofa function of at least one representative value of at least one randomvariable, comprising: a step of obtaining the at least one randomvariable; a step of determining the at least one representative value ofthe obtained at least one random variable; a step of determining astatistic of the obtained at least one random variable; a step ofdetermining a gradient of the function with respect to the determined atleast one representative value; and a step of transforming the obtainedstatistic of the at least one random variable into a statistic of thefunction, using the determined gradient.
 2. The method according toclaim 1, wherein the step of transforming comprises transforming thestatistic of the at least one random variable into the statistic of thefunction, such that the statistic of the function responds to thestatistic of the at least one random variable more sensitively in thecase of the gradient being steep than in the case of the gradient beinggentle.
 3. The method according to claim 1, wherein each one of therepresentative value of the each random variable comprises at least oneof an average, an arithmetic mean, a geometric mean, a median, aharmonic mean, and a mode, of each one of the at least one randomvariable.
 4. The method according to claim 1, wherein the step ofdetermining the representative value comprises determining the at leastone representative value of the at least one random variable upontruncating at least one part of individual data values belonging to theat least one random variable.
 5. The method according to claim 1,wherein the statistic of each one of the at least one random variablecomprises at least one of a standard deviation, a confidence interval, aset of data, a probability density function, and a cumulative densityfunction, of the each random variable.
 6. The method according to claim1, wherein the statistic of the function comprises at least one of astandard deviation, a confidence interval, a set of data, a probabilitydensity function, and a cumulative density function, of the function. 7.The method according to claim 1, further comprising a step of estimatingthe measure of randomness of the function of the at least onerepresentative value, on the basis of the statistic of the function. 8.The method according to claim 7, wherein the measure of randomnesscomprises a range of a confidence interval of the function of the atleast one representative value.
 9. The method according to claim 7,applied to a simulation for discrete event, results of which simulationis used to effect the method, wherein the step of estimating comprisesestimating the measure of randomness using results of only one executionof the simulation.
 10. The method according to claim 9, wherein anaccuracy to be satisfied with the statistic of the function ispredetermined, and the step of determining a statistic comprises: (a)determining the statistic of the at least one random variable, on thebasis of a sum of individual data values belonging to the at least onerandom variable; (b) determining the statistic of the at least onerandom variable on the basis of the sum, upon adding to the sum at leastone new individual data value belonging to the at least one randomvariable; (c) determining the statistic of the at least one randomvariable when at least one new individual data belonging to the at leastone random variable becomes available during the simulation; (d)transforming the determined statistic of the at least one randomvariable into the statistic of the function; and (e) terminating thesimulation when the predetermined accuracy is satisfied with thestatistic of the function.
 11. The method according to claim 1, whereinthe function is a function of a plurality of random variables, the stepof transforming comprising: (a) determining a measure of randomness ofeach one of the plurality of random variables at or in the vicinity of arepresentative value of each one of the obtained plurality of randomvariables, as the statistic of each random variable; (b) determining ameasure of dependence between the plurality of random variables; and (c)transforming the determined measures of randomness of the plurality ofrandom variables into a measure of randomness of the function, using thedetermined measure of dependence and the determined gradient.
 12. Themethod according to claim 11, wherein the measure of randomness of theeach random variables comprises at least one of a maximum likelihoodestimator of a variance of the each random variable, an unbiasedestimator of the variance, a maximum likelihood estimator of a standarddeviation of the each random variables, an unbiased estimator of thestandard deviation, a variance of a representative values of the eachrandom variable, a standard deviation of a representative value of theeach random variable, a coefficient of variation of the each randomvariable, a general central moment of the each random variable, aconfidence interval of the each random variable, a set of dataindicative of the each random variable, a probability density functionof the each random variable, and a cumulative density function of theeach random variable.
 13. The method according to claim 11, wherein themeasure of dependence comprises at least one of an unbiased estimator ofa covariance of the plurality of random variables, a maximum likelihoodestimator of the covariance, and a correlation coefficient of theplurality of random variables.
 14. The method according to claim 1,wherein the function is a function of a plurality of random variables,the step of transforming comprises transforming the obtained statisticof the plurality of random variables into the statistic of the function,without a calculation of a measure of dependence between the pluralityof random variables.
 15. A method of determining a set of data of afunction of a representative value of each one of at least one randomvariable, which set of data allows an evaluation of a statistic of thefunction, comprising: a step of obtaining a set of individual datavalues belonging to each random variable, which set represents anapproximation of a distribution of the each random variable; a step ofdetermining the representative value of the each random variable; a stepof determining a gradient of the function with respect to the determinedrepresentative value; and a step of transforming the obtained set ofindividual data values into the set of data representing the function.16. The method according to claim 15, wherein the step of transformingthe set of individual data values of the each random variable into theset of data representing the function, such that the set of datarepresenting the function responds to the set of individual data valuesmore sensitively in the case of the gradient being steep than in thecase of the gradient being gentle.
 17. The method according to claim 15,further comprising a step of estimating a measure of randomness of thefunction of the representative value, on the basis of the set of datarepresenting the function.
 18. The method according to claim 17, whereinthe measure of randomness comprises a range of a confidence interval ofthe function of the representative value.
 19. The method according toclaim 17, applied to a simulation for discrete event, results of whichsimulation is used to effect the method, wherein the step of estimatingcomprises estimating the measure of randomness using results of only oneexecution of the simulation.
 20. The method according to claim 1,applied to an analysis of a plurality of business models to be acceptedin realizing a given business, wherein a function of at least one ofrandom variable is predetermined for each one of the plurality ofbusiness models, and the function of a representative value of the eachrandom variable for one of the plurality of business models is to becompared with the function of a representative value of the each randomvariable for another of the plurality of business models.
 21. A methodof estimating a measure of randomness of at least one random variable tosatisfy a predetermined condition regarding a measure of randomness of afunction of at least one representative value of the at least one randomvariable, the predetermined condition being formulated to define acentral location and a measure of dispersion, of a distribution of thefunction, comprising: a step of determining a gradient of the functionwith respect to the defined central location; and a step of determiningthe measure of randomness of the at least one random variable, on thebasis of the determined gradient and the defined measure of dispersion.22. The method according to claim 21, wherein the step of determiningthe measure comprises transforming the defined measure of dispersioninto the measure of randomness of the at least one random variable, suchthat the measure of randomness of the at least one random variableresponds to the defined measure of dispersion more sensitively in thecase of the gradient being steep than in the case of the gradient beinggentle.
 23. The method according to claim 21, wherein the measure ofdispersion comprises at least one of a standard deviation, a confidenceinterval, a set of data, a probability density function, and acumulative density function, of the function.
 24. The method accordingto claim 21, wherein the measure of randomness of each one of the atleast one random variable comprises at least one of a standarddeviation, a confidence interval, a set of data, a probability densityfunction, and a cumulative density function, of the each randomvariable.
 25. A computer program to be executed by a computer to effectthe method according to claim
 1. 26. A computer-readable storage mediumhaving stored therein the computer program according to claim 25.